ENGI 441 Cetral Limit Therem Page 11-01 Cetral Limit Therem [Navidi, secti 4.11; Devre sectis 5.3-5.4] If X i is t rmally distributed, but E X i, V X i ad is large (apprximately 30 r mre), the, t a gd apprximati, X ~ N, At "http://www.egr.mu.ca/~ggerge/441/dems/clt.html" is a demstrati web prgram t illustrate hw the sample mea appraches a rmal distributi eve fr highly -rmal discrete distributis f X. Csider the expetial distributi, whse p.d.f. (prbability desity fucti) is x 1 1 f x; e, x 0, 0 E X, VX It ca be shw that the exact p.d.f. f the sample mea fr sample size is 1 x x e f x;,, x 0, 0, X 1! 1 1 with mea ad variace E X, V X. [A -examiable derivati f this p.d.f. is available at " http://www.egr.mu.ca/~ggerge/441/dems/cltexp.pdf ".] Fr illustrati, settig = 1, the p.d.f. fr the sample mea fr sample sizes = 1,, 4 ad 8 are: 1: f x e x x : f x 4xe 3 4x 44x e 4: 8: f x f x X 3! X X 7 8 x e 88 7! x The ppulati mea = E[X] = 1 fr all sample sizes. The variace ad the psitive skew bth dimiish with icreasig sample size. The mde ad the media apprach the mea frm the left.
ENGI 441 Cetral Limit Therem Page 11-0 Fr a sample size f = 16, the sample mea X has the p.d.f. f X x 15 16 x e x 16 16 1 ad parameters E X 1 ad V X 15!. 16 A plt f the exact p.d.f is draw here, tgether with the rmal distributi that has the same mea ad variace. The apprach t rmality is clear. Beyd = 40 r s, the differece betwee the exact p.d.f. ad the Nrmal apprximati is egligible. It is geerally the case that, whatever the prbability distributi f a radm quatity may be, the prbability distributi f the sample mea X appraches rmality as the sample size icreases. Fr mst prbability distributis f practical iterest, the rmal apprximati becmes very gd beyd a sample size f 30. Example 11.01 A radm sample f 100 items is draw frm a expetial distributi with parameter = 0.04. Fid the prbabilities that (a) a sigle item has a value f mre tha 30; (b) the sample mea has a value f mre tha 30. (a)
ENGI 441 Cetral Limit Therem Page 11-03 Example 11.01 (ctiued) (b)
ENGI 441 Cetral Limit Therem Page 11-04 Sample Prprtis A Berulli radm quatity has tw pssible utcmes: x = 0 (= failure ) with prbability q1 p ad x = 1 (= success ) with prbability p. Suppse that all elemets f the set X 1, X, X 3,, X are idepedet Berulli radm quatities, (s that the set frms a radm sample). Let T X 1 X X 3 X umber f successes i the radm sample ad Pˆ T = prprti f successes i the radm sample, the T is
ENGI 441 Cetral Limit Therem Page 11-05 Example 11.0 55% f all custmers prefer brad A. Fid the prbability that a majrity i a radm sample f 100 custmers des t prefer brad A.
ENGI 441 Cfidece Itervals (Oe Sample) Page 11-06 Cfidece Itervals (Oe Sample) [Navidi, sectis 5.1-5.3; Devre chapter 7] S far we have cstructed prbability statemets hw likely certai sample values are, give kwledge f the ppulati frm which the radm sample came. Nw we shall reverse that situati: we have a kw sample i frt f us, frm which we ca ifer the values f the parameters f the ppulati frm which the sample was draw. This is the realm f iferetial statistics. If the radm quatity X is such that ~ N, X, the it is highly ulikely that X will be mre tha three stadard deviatis away frm its mea: P X 3 PX 3 PZ 3 3.00.003 Mre tha 99.7% f the time, X will be clser tha three stadard deviatis t its mea. Fr a sufficietly large radm sample, the cetral limit therem assures us that the sample mea X ~ N, (either exactly r t a excellet apprximati). Therefre we have 99.7% cfidece that a bserved sample mea x is withi three stadard errrs f the ppulati mea. This lie f reasig allws us t replace a pit estimate by a rage f plausible values f a ukw parameter a cfidece iterval. Mre geerally, whe X ~ N,, P z/ X z/ 1
ENGI 441 Cfidece Itervals (Oe Sample) Page 11-07 The cfidece iterval estimatr fr (at a level f cfidece f 1 ) is X z X z The 1 cfidece iterval estimatr fr is a radm iterval X z, X z The prbability is (1 ) that the abve radm iterval icludes the true value f.. 1 1 cfidece f all radm samples will prduce a iequality, (the iterval estimate fr ) x z x z that is true. Nte that the cfidece iterval estimate ctais radm quatities at all! The statemet is either abslutely certai t be true r abslutely certai t be false, (depedig the values f,, x, ad ). Iterpretati f a cfidece iterval [ = cfidece iterval estimate ] Oly 5% f all 95% cfidece iterval estimates fr fail t iclude.
ENGI 441 Cfidece Itervals (Oe Sample) Page 11-08 A ccise expressi fr the C.I. (cfidece iterval estimate fr ) is x z A Bayesia view f iterval estimati: If the ly quatity amg {,, x, ad } that we d t kw is, the represet the ukw by the radm quatity A. The A te abut the stadard rmal distributi ad the t distributi Let Z ~ N(0, 1) (stadard rmal distributi), s that P Z z z (cumulative distributi fucti fr the stadard rmal distributi). th The the 1 100 percetile f the stadard rmal distributi is z, which satisfies PZ z. It als fllws that 1 z z. 1 z The t distributi with ν degrees f freedm is als a bell shaped curve, with a mea, media ad mde at t 0, but with a greater variace tha the stadard rmal distributi. As the umber f degrees f freedm icreases, the t distributi appraches the z (stadard rmal) distributi. The graphs f t 1 ad t 5 are shw here, tgether with z, which is idistiguishable t the eye frm t fr abve 30 r s. Therefre lim t, t, z. Use the t distributi ly if the true ppulati variace is ukw.
ENGI 441 Cfidece Itervals (Oe Sample) Page 11-09 1 100 percetile z, use the fial rw i the table f critical values T fid the th f the t distributi ( page 17-0 r the iside back cver f the textbk): z t., The fial rw f the t tables is 0.1 0.05 0.05 0.01 0.005 1.8155 1.64485 1.95996.3635.57583 Therefre PZ 1.645.05 r equivaletly z.05 1.645 ; P 1.960.05 Z r equivaletly z.05 1.960 ; etc. Example 11.03 The rate f eergy lss X (watt) i a mtr is kw t be a rmally distributed radm quatity with stadard deviati 3.0 W. A radm sample f 100 such mtrs prduces a sample mea rate f eergy lss f 58.3 W. Fid a 99% cfidece iterval estimate fr the true mea rate f eergy lss.
ENGI 441 Cfidece Itervals (Oe Sample) Page 11-10 Example 11.03 (ctiued) Hw large must be fr the width f the 99% cfidece iterval estimate fr t be less tha 1.0? Chice f sample size The width f the cfidece iterval x z, x z is w z z w The sample size is iversely related t the square f the desired width. Edpits f a (1 ) CI fr : (a) kw: (b) ukw, large: (c) ukw, small: Whe is small, X must be early (r exactly) rmal.
ENGI 441 Cfidece Itervals (Oe Sample) Page 11-11 Example 11.04 The lifetime X f a particular brad f filamets is kw t be rmally distributed. A radm sample f six filamets is tested t destructi ad they are fud t last fr a average f 1,008 hurs with a sample stadard deviati f 6. hurs. (a) Fid a 95% cfidece iterval estimate fr the ppulati mea lifetime. (b) Is the evidece csistet with 1000? (c) Is the evidece csistet with > 1000?
ENGI 441 Cfidece Itervals (Oe Sample) Page 11-1 Prperties f a cfidece iterval If we thik f the width f the cfidece iterval as specifyig its precisi, the the cfidece level (r reliability) f the iterval is iversely related t its precisi. Estimati f Ppulati Prprti Whe a radm sample f size is draw frm a ppulati i which a prprti p f the items are successes, the, as we saw page 11.04, pq P ˆ ~ N p, fr sufficietly large p ad q, (amely, p > 10 ad q > 10 ). Cmpare this with X ~ N,, fr which the crrespdig cfidece iterval has edpits x z / s. Hwever, the variace pq is ukw because p ad q are ukw. A bvius remedy is t replace the ukw parameters p ad q by their pit estimates ˆp ad ˆq.
ENGI 441 Cfidece Itervals (Oe Sample) Page 11-13 Therefre, a simple 100(1 )% cfidece iterval estimatr fr p is P z / PQ ad the 100(1 )% cfidece iterval estimate fr p is pˆ z / pq ˆˆ Hwever, these cfidece itervals ca exhibit sigificat errrs whe either p r q is much less tha 100. Durig the 1990 s, mre reliable cfidece itervals fr p were develped. Oe f them is (Devre, sixth editi, secti 7., page 66) : z pq ˆˆ z pˆ 4 z / 1 / / z / Ather iterval, frm the Navidi textbk (page 339), is the Agresti-Cull iterval. If x is the bserved umber f successes i a radm sample f idepedet Berulli trials, the defie x* x ad * 4 s that p* x* x * 4 ad q* 1 p* The the 100(1 )% cfidece iterval estimate fr p is p* z / p* q* * It turs ut that the Bayesia pit estimate fr p is p*, t ˆp.
ENGI 441 Cfidece Itervals (Oe Sample) Page 11-14 Example 11.05 Frm a radm sample f e thusad silic wafers, 750 pass a quality ctrl test. Fid a 99% cfidece iterval estimate fr p (the true prprti f wafers i the ppulati that are gd). = 1000 ad x = 750 x 750 3 p ˆ 1000 4 1 q ˆ 1 pˆ 4.005 z.005 t.005,.576 Edpits f the C.I.: pq ˆˆ.75.5 pˆ z.75.576 =.75.035 7... 1000 Therefre the 99% cfidece iterval estimate fr p is crrect t three sigificat figures. 71.5% p 78.5% Usig the mre precise Agresti-Cull versi f the cfidece iterval yields x* = 750 + = 75, * = 1000 + 4 = 1004 p* x* 75 * 1004.749003 p* q*.749.50 * 1004.013683 The 99% CI is therefre p* q* p* z.005.749 *.576.0136.749003.03547 71.4% p 78.4%
ENGI 441 Cfidece Itervals (Oe Sample) Page 11-15 (1)100% Bayesia Cfidece Iterval fr [t i the Navidi r Devre textbks] Suppse that previus evidece leads us t believe that. The stregth f this belief is represeted by the variace (lwer variace crrespds t strger belief). We wish t update that estimate after a radm sample f size has bee examied. Assume that 30 (s that the Cetral Limit Therem will apply). Prir distributi: New evidece: X ~ N, Sample size = Sample mea = x Sample stadard deviati = s Calculate where wd, w x d w d w w 1, w w w are the weights f the data ad rigial ifrmati respectively, give by wd 1 1, w s d Psterir distributi: X ~ N, 1 100% Bayesia iterval fr : z / Cmpare with the classical 1 100% cfidece iterval fr : s x z / ( 30) r x z / I may applicatis, the Bayesia iterval is fte arrwer tha the classical cfidece iterval, because the Bayesia iterval icrprates mre ifrmati (previus evidece r belief abut the true value f ). [Nte: it is easy t shw that as (r if x the), * = x ad that as, * s, which are the classical expressis.]
ENGI 441 Cfidece Itervals (Oe Sample) Page 11-16 Examples f Bayesia Cfidece Itervals These examples are mdificatis f the previus examples f classical cfidece itervals fr. Example 11.06 (mdificati f Example 11.03) The rate f eergy lss X (watt) i a mtr is kw t be a rmally distributed radm quatity ad prir experiece suggests that the mea is 60 W with stadard deviati 3.0 W. A radm sample f 100 such mtrs prduces a sample mea rate f eergy lss f 58.3 W with sample stadard deviati.8 W. Fid a 99% cfidece iterval estimate fr the true mea rate f eergy lss.
ENGI 441 Cfidece Itervals (Oe Sample) Page 11-17 Example 11.07 (mdificati f Example 11.04) The lifetime X f a particular brad f filamets is kw t be rmally distributed. Prir experiece suggests that 1000 ad 6.0. A radm sample f six filamets is tested t destructi ad they are fud t last fr a average f 1,008 hurs with a sample stadard deviati f 6. hurs. (a) Fid a 95% cfidece iterval estimate fr the ppulati mea lifetime. (b) Is the evidece csistet with 1000? [Ed f Chapter 11]
ENGI 441 Cfidece Itervals (Oe Sample) Page 11-18 [Space fr Additial Ntes]